At present, a passive element such as a Surface Acoustic Wave (SAW) filter is mainly used as a filer circuit for Radio Frequency (RF) signals. In such filter circuits, the output signal does not have an output gain, thereby causing noise degradations of signals. Since the SAW filters have a narrow adaptive frequency domain, multiple filter circuits are needed to cover a wide bandwidth.
As a filter for eliminating the above drawback of the SAW filters, there is a recursive filter circuit, for example. The recursive filter circuit is also called positive feedback filter circuit, includes positive feedback therein, has a wide adaptive frequency domain, and is capable of covering a wide bandwidth.
FIG. 7 is a view for showing a conventional recursive filter circuit. The conventional recursive filter circuit is composed of: a Low Noise Amplifier (LNA) 201, a variable gain amplifier 204, a phase shifter 203, and an adder 202. In the circuit shown in FIG. 7, when the amplitude and phase of an output signal Vout from the adder 202 are substantially same with the amplitude and phase of a feedback signal Vf from the variable gain amplifier 204 and the phase shifter 203, this circuit obtains a large gain at a certain frequency and functions as a filter circuit.
The control of the recursive filter circuit shown in FIG. 7 will now be described.
The transfer function of the recursive filter circuit shown in FIG. 7 is given by the following expression (1).H=GL/(1−Gg|K|exp(j∠K))  expression (1)
In the above expression (1), GL represents a gain of the LNA 20, Gg represents a gain of the variable gain amplifier 204, and K represents a transfer function of the phase shifter 203. The gain of the adder 202 is substantially 1.
When the denominator of the expression (1) is substantially 0, the recursive filter circuit of FIG. 7 has a large gain, sharp peak characteristics, and steep frequency characteristics, which are observed by the expression (1), In this situation, the following expressions (2) and (3) are satisfied.|GgK|≈1  expression (2)∠[Gg|K|exp(j∠ZK)]≈2π  expression (3)
where in the real part of the expression (1), when Re[Gg|K|exp(j∠K)]>1 is satisfied, the filter circuit oscillates.
By making the frequency (center frequency) in a state of immediately before the oscillation, the recursive filter circuit functions properly as a filter.
With the above conditions, there are three following conditions for controlling the recursive filter circuit to operate properly as a filter.
That is, in consideration of a closed-loop circuit composed of the variable gain amplifier, the phase shifter, and the adder, the following conditions can be extracted from |GgK|≈1.                The amplitude of the output signal Vout and that of the feedback signal Vf are substantially same with each other.        The loop gain |GgK| is made slightly smaller than 1.        
Likewise, in consideration of a closed-loop circuit composed of the variable gain amplifier, the phase shifter, and the adder, the following conditions can be extracted from ∠[Gg|K|exp(j∠K)]≈2π.                The phase of the output signal Vout and that of the feedback signal Vf are substantially same with each other.        